(09/13/2014, 07:15 PM)sheldonison Wrote: \( \exp(x)\sqrt{x} \approx \sum_{n=0}^{\infty} \frac{x^n}{\Gamma(n+0.5)}\;\; \) Have you seen this excellent asymptotic entire series?
Yes Ive seen this before.
Have you heard of hypergeometric functions or generalized hypergeometric functions ?
When I was still active at sci.math I mentioned the underrated and insuffiently known properties and powers of the hypergeometric functions and more particular their INVERSES.
Inverse hypergeometric functions is one those other " crazy math concepts " apart from tetration , collatz and prime twins.
Bill Dubuque and a few others recognized / remembered the power of these functions.
But its far from mainstraim.
It is amazing how unexpectedly these inverse hypergeometric functions can occur.
Literature is very very rare ( like tetration ) and it is not introduced to younger students.
From " entire function theory " comes the idea that hidden recursions
are often of hypergeometric nature.
hypergeometric and their inverse occur in for instance closed form solutions ( with integral or sum ) for half-iterates.
They are also very good approximations to other functions.
A good understanding of the gamma function is essential.
Id say Euler and Gauss were the first to recognise its importance , but that is open for debate.
The theory is very uncomplete ( as for tetration I guess ) , and that might be the reason for the avoidance in more classical math.
That's enough background.
I have to think about this , I have seen it before ...
But that is probably over 10 years ago and I have to dig in my memory.
Maybe its easy , but I lack time and concentration now.
Perhaps this matters or helps :
http://www.math.upenn.edu/~wilf/AeqB.pdf
edit :
this appears as the fake sqrt (after division by exp).
I conjecture for x > 2 : \( |\exp(x)\sqrt{x} - \sum_{n=0}^{\infty} \frac{x^n}{\Gamma(n+0.5)}\;\;| < sqrt {x+1} \)
regards
tommy1729
